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Is there any simple graph $\Gamma$ with 16 vertices with full automorphism group $G$ such that $H\cong Q_8$ be a semiregular normal subgroup of $G$?

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$Q_8$ has outer automorphisms of order 2. So you can extend $Q_8$ using such an automorphism, obtaining $G$ of order 16... I hope it is not your homework. :) –  Dima Pasechnik Dec 25 '12 at 17:07
Thanks for your answer. Indeed I want to be $G$ as full automorphism group of a graph with 16 vertices. So I change my question. –  majid arezoomand Dec 25 '12 at 19:45
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